House Republicans continued with the piecemeal government funding approach Thursday, even as chatter turned to whether Speaker John A. Boehner could finally pull off a "grand bargain" on government spending and the debt limit.

But later Wednesday night, the speaker met with President Barack Obama, House Minority Leader Nancy Pelosi, Senate Majority Leader Harry Reid and Senate Minority Leader Mitch McConnell at the White House. And the negotiations — or lack thereof — seemed to have left a sour taste in the mouth of Republicans.

“The president reiterated one more time he will not negotiate,” Boehner said just after the White House meeting.

While it’s clear leaders were unable to immediately find a solution to reopen the government, the meeting, which lasted more than an hour, covered the CR and the debt limit, which expires Oct. 17. National Review's Robert Costa also reported that Boehner brought up the idea of a "grand bargain" at the White House Wednesday evening.

Costa quoted a Democratic source as saying:

"Boehner raised the prospect of a grand bargain-type deal at the White House meeting and was laughed at because everyone feels like they’ve heard this song and dance before. The general feeling is, if he’s really ready to make some tough choices — read, revenue — then great. But the history of this from where we sit is Boehner talking a big game, then bailing as soon as he runs into the inevitable resistance from a certain faction in his caucus."

This time could be different — if Boehner is willing to buck the tea party wing of the House GOP conference. On Thursday, The New York Times reported that Boehner has assured some GOP lawmakers he will not allow the U.S. to default on its debt — a scenario that would likely have catastrophic consequences for U.S. economy and the world.

As The New York Times reported, one Republican lawmaker said Boehner has said he would allow a debt ceiling increase to pass with a majority of Democratic votes:

"The lawmaker, who spoke on the condition of not being named, said Mr. Boehner indicated he would be willing to violate the so-called Hastert rule if necessary to pass a debt limit increase. The informal rule refers to a policy of not bringing to the floor any measure that does not have a majority of Republican votes."

Also on Thursday, Cantor criticized the president for his "my way or the highway" stance.

“We should sit down and have a discussion,” the Virginia Republican told reporters at a morning news conference. “But because there is an insistence on no negotiations, no talking, my way or the highway, we are here.”

While walking down a hallway, Cantor said he found it “unbelievable” that Obama “would call Speaker Boehner and others over to the White House just to let them know that he wouldn’t negotiate.”

Despite apparent behind the scenes maneuvering, the House voted 265-160 on a GOP bill that would pay military reserve personnel during the shutdown. It’s the fourth mini-continuing-resolution the House has passed, in addition to a bill that was signed by Obama to pay active duty military personnel.

While Democrats assented to military pay bill, they appear resolute in not allowing Republicans to fund only the popular pieces of the government in an effort to mitigate the effects of a shutdown — in reality and politically.

The ranking Democrat of the House Armed Services Committee, Rep. Adam Smith of Washington, summed up why Democrats agreed to paying the military during a shutdown, but not funding other important pieces of government.